We report on the first measurement of the fission barrier height in a heavy shell-stabilized nucleus. The fission barrier height of 254No is measured to be Bf=6.0±0.5 MeV at spin 15ℏ and, by extrapolation, Bf=6.6±0.9 MeV at spin 0ℏ. This information is deduced from the measured distribution of entry points in the excitation energy versus spin plane. The same measurement is performed for 220Th and only a lower limit of the fission barrier height can be determined: Bf(I)>8 MeV. Comparisons with theoretical fission barriers test theories that predict properties of superheavy elements.